Mathematical Sciences: Numerical Methods for Large-Scale Ocean Modeling

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Higdon The investigator studies the numerical solution of partial differential equations that describe large-scale ocean circulation. The main topic of investigation is operator splitting for the governing equations. These equations admit wave velocities having widely varying magnitudes, and this situation creates serious problems with the stability and efficiency of numerical algorithms. One common approach to resolving these problems is to split the fast and slow dynamics into separate subproblems. The fast motions are essentially independent of depth, and it is natural to try to model these motions with a two-dimensional system of equations. The slow motions would then be modeled with a three-dimensional system. However, if the splitting is inexact, then the equations that supposedly model the slow motions may actually admit fast motions, to some degree. There then arises the possibility that these fast motions might cause numerical instabilities in the computational algorithm. The main goal of this project is to develop a stable, efficient, and accurate operator splitting for the governing equations. Secondary goals are to develop open boundary conditions for limited-area models and to examine higher-order difference methods for spatial discretizations. One of the major scientific challenges of the present day is to obtain high-resolution computer simulations of global ocean circulation over periods of many years. Such simulations are of great interest because the ocean is a crucial component of the earth's climate system, so studies of possible global change must include a careful examination of the role of the ocean. For example, the ocean has a great capacity to transport heat energy; part of this transport is due to bulk transport by major current systems, and part is due to mixing caused by smaller-scale eddies that are shed from currents. Comprehensive data on these processes are difficult to collect, and it appears that an accurate understanding of the earth's climate system will require numerical simulations based on an eddy-resolving model of the general circulation of the ocean. Such simulations will present great computational demands. The goal of this project is to contribute to the development of suitable numerical methods for these computations. This work is being performed in collaboration with members of the physical oceanography modeling group in the College of Atmospheric and Oceanic Sciences at Oregon State University.